#10324144

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,16)(12,18)(13,19)(14,21)(15,17)(20,24)(22,23)(25,26)(27,33)(28,36)(29,31)(30,35)(32,38)(34,39)(37,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,13)(10,16,17,19)(12,20,23,21)(14,18,24,22)(25,27,29,35)(26,30,31,33)(28,37,38,34)(32,40,36,39);(2,5)(6,7)(10,17)(11,13)(18,22)(20,21)(26,31)(30,33)(32,36)(39,40);(2,5)(6,7)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,19,24,16)(25,28,29,38)(26,32,31,36)(27,34,35,37)(30,39,33,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,19,17,16)(12,20,23,21)(14,18,24,22)(25,26)(27,30)(28,32)(29,31)(33,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,19,23,16)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,40)(37,38)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10231340);